Regular Forward

• You agree now to buy an asset at time t and the money is exchanged at time t. This equals what we expect the stock to be worth at time t.

• \(F=S_0 e^{(r-\delta)t}\)
  • Continuous Dividends
  • Compound by r, discount by \(\delta\) b/c you aren’t receiving the dividends
  • Remember to always compound by r, even if you are given the expected return of the stock (\(\alpha\))
• \(F=[S_0 - \sum (Div_i)(e^{-rt_i})]e^{rt}\)
  • Discrete Dividends
  • Discount each dividend to the PV so you get them all at the same time, then compound everything to time t. We use the rfr for all discounting and compounding here.

Prepaid Forward

• You pay upfront to receive the asset at time t. Asset holder still receives dividends during this time.

• \(F^P = S_0e^{-\delta t}\)
  • Continuous dividends
  • Just discount the stockprice by the dividends b/c you don’t get these
• \(F^P = S_0 - \sum (Div_i)(e^{-rt_i})\)
  • Discrete dividends are discounted by the continuous rfr rate

Symbols

• \(F\): Forward price you pay at time t
• \(F^P\): Prepaid forward price you pay at time 0
• \(S_0\) or just \(S\): Stock price at t=0
• \(r\): Risk free interest rate
• \(\delta\): continuously compounded dividend rate
• \(t\): time until expiration

Relationship between regular and prepaid forwards

• \(F = F^Pe^{rt}\)
• Regular forwards equations are just the prepaid forward equations compounded by \(e^{rt}\)

Currency Forwards

• Think of currency forwards just like asset forwards. But that you are receiving a foreign currency as the asset. The symbols and language is different but you can still use the above formulas.
• \(F = x_0e^{(r-r_f)t}\)
• \(F^P = x_0e^{-r_f(t)}\)
  • \(x_0\) = Exchange Rate, how much you have to pay in base currency to receive the foreign currency. (\(S_0\))
  • \(r\) = rfr for the base currency you currently have
  • \(r_f\) = rfr for the foreign currency you are buying (\(\delta\))

Futures

• Futures use the same formulas as forwards
• Futures are traded on exchanges while forwards are traded over the counter
• Futures are cash settled, and deal with margins
• Some language
  • Notional Value = Index * Size
  • Margin = %age of notional value that you have to keep in your margin account (in case you default)
  • Maintenance Margin = %age of your original margin balance that is the minimum allowed amount. You will get a marign call if you have less money than the maintenance margin and have to deposit more money.
• Ex: You enter into a futures contract to long the S&P 500. The size is $200, the index is 700, margin is 25%, maintenance margin is 80%, and risk free rate is .05.
  • Balances at t=0:
    • Notional Value = Index * Size = \(\$700(200) = \$140,000\)
    • Initial balance in the margin account = (Notional Value)(margin %) = \(\$140,000(.25) = \$35,000\)
    • Minimum balance before a margin call = \(\$35,000(.8) = \$28,000\)
  • 1 week passes. Index price rises to 730.
    • Margin account balance grows by rfr for 1 week: \(\$35,000e^{.05(\frac{1}{52})}=35033.67\)
    • Index price rose by \(730-700=30\), so margin balance increases by \(30*200=6,000\)
    • Margin balance at t = 1 week:
      • \(=\$35,000e^{.05(\frac{1}{52})} + 6,000=41033.67\)
  • Another week passes. t = 2 weeks. Index price falls to $650.
    • Margin grows by rfr, and margin balance decreases by \(730-650=\$80\) times the size of 200
    • Margin Balance at t = 2 weeks \(=\$4.103367\times 10^{4}e^{(\frac{.05}{52})} - (80)(200)=25073.14\)
    • So we would get a margin call to add more money to the margin account since the value dropped below the maintenance margin of $28,000
• If you are short the index, then when the index rises this is a bad thing, and your margin account should decrease.
  • For ex, take the week 1 scenario above when the index rises to 730. Instead of growing the margin balance by \(30*200=6,000\), we would decrease the margin balance by $6k.

Overview

• Main types
  • European = can only be exercised at expiration, easier math so assume this
  • American = can be exercised at any point prior to expiration, must use binomial trees for pricing this
• Exotic Types
  • Asian, Barrier, Compound, Gap, Exchange
• Payoffs
  • Long Call = max{0, Spot - Strike}
    • Buying the option to buy the stock at the strike price
    • Hoping stock goes up, so you can buy at low strike, and sell at high spot
  • Long Put = max{0, Strike - Spot}
    • Buying the option to sell the stock at the strike price
    • Hoping stock goes down, so you can buy the stock at the low spot, and sell at a high strike
  • Short Call = min{0, Strike - Spot}
    • Selling someone the option to buy the stock from you at the strike price
    • Hoping the stock will go down so the option you sold is never exercised w/ 0 payoff. So you just profit the premium.
  • Short Put = min{0, Spot - Strike}
    • Selling someone the option to sell you the stock at the strike price
    • Hoping the stock goes up, so the spot > strike, so they won’t sell it to you at the lower price. So you just profit the premium.
• Profits
  • = payoff + FV of the premium (compounded by the rfr)
• Language
  • Spot price: actual price of the stock at expiration
  • Strike: agreed upon price that will be exchanged for the asset

Options Profit Plots

Arbitrage

Basic Profit Functions

• Profit Long Call = \(Max[0, S_T - K] - FV(Prem)\)
• Profit Short Call = \(-Max[0, S_T - K] + FV(Prem)\)
• Profit Long Put = \(Max[0, K - S_T] - FV(Prem)\)

• Profit Short Put = \(-Max[0, K-S_T]+FV(Prem)\)

  • Short & Long of each are just multiplied by -1
  • Calls are \(S_T-K\), and Puts are \(K-S_T\)

Put Call Parity

• This parity must exist between the Call and Put premium for the same asset with the same strikes and expiration dates, or else arbitrage exists.

• For Stocks

  • \(C - P = Se^{-\delta t} - Ke^{-rt}\)
  • \(C - P + Ke^{-rt} = Se^{-\delta t}\)
    • Can rearrange so it looks like this. The left side of the equation is a synthetic prepaid forward, the right side of the equation is the actual prepaid forward.

• For currency

  • \(C-P = x_0e^{-r_f t}-Ke^{-rt}\)
• Symbols
  • C = Call Premium
  • P = Put Premium
  • S = \(S_0\) Stock price at t=0
  • \(x_0\) = Foreign currency price at t=0
  • \(\delta\) = dividend yield of stock
  • \(r_f\) = rfr of foreign currency (\(\delta\))
  • Just use 1st equation and treat foreign currency like the stock

Binomial Trees

• AKA “The Usual Method in McDonald”, “The Standard Method”, “A Tree based on forward prices”

1 Period Binomial Tree

• Assume stock can either rise from \(S_0\) to \(S_u\), or go down to \(S_d\) at time t.

• Option Premium = \(\Delta S + B\)
  • \(\Delta = e^{-\delta t} \frac{V_u - V_d}{S(u - d)}\)
  • \(B=e^{-rt} \frac{u V_d - d V_u}{u-d}\)
    • \(V_u/V_d\) = Option payoff if stock rises/drops. For call or put
    • \(S\) = \(S_0\) = Stock price at t=0
    • \(u\) & \(d\) = Ratio of increase or decrease
      • \(u = \frac{S_u}{S_0}\)
      • \(d = \frac{S_d}{S_0}\)

  • For calls we’ll get a \(+\Delta\) & \(-B\). For puts we’ll get a \(-\Delta\) and \(+B\)

  • Synthetic position
    • Can use this formula to create an equivalent synthetic position
    • Call payoff = purchasing \(+\Delta\) shares of stock, and borrowing \(-B\) to do so
    • Put payoff = shorting \(-\Delta\) shares of stock, and lending \(+B\)

• Arbitrage

  • If the Option Prem \(\neq\) \(\Delta S + B\), then we buy what’s lower, and short what’s higher
  • Profit = \(|C^* - C|e^{rt}\)
    • Difference between what premium actually is and what it should be
    • Profit is calculated at time t=t

Black Scholes

• This is how modern options are priced today.
• This framework assumes:
  • Continuously compounded Stock Returns are Normally Distributed
  • The ratios of the change of stock prices (\(\frac{S_t}{S_0}\)) are Lognormally distributed
    • And thus, the log ratios of the change in stock prices (\(ln(\frac{S_t}{S_0})\)) are Normally distributed
  • No Transaction costs
  • Volatility and Risk free rate are constant

Greeks of multiple positions

• \(\Delta = \sum_{i=1}^{N} (n_i)(greek_i)\)

  • For a portfolio of options, you just sum up all the \(\Delta\)’s or \(\Gamma\)’s or any Greek of each option, to get the net Greek value.
  • When long, \(\Delta\) => 1. When Short => \(\Delta\) => -1.

Other Greek-ish formulas to memorize

• Option Elasticity (\(\Omega\))

  • \(\Omega = \frac{S\Delta}{V}\) = \(\frac{(V_t-V_0)/V_0}{(S_t-S_0)/S_0}\) = (% Change in Option price)/(% Change in Stock price)
    • \(\Omega\) is the 3rd derivative of the options price, or derivative of \(\Gamma\)
    • \(\Delta\) = Greek Delta of position (or net \(\Delta\) of portfolio)
    • V = Option premium for Call or Put, (or net premium of portfolio)
    • S = Current stock price
    • Represents: If the stock changes by 1%, by what % does the option change?
    • To remember: equation looks like SUD

  • Portfolio Elasticity

    • \(\Omega_{Portfolio} = \sum \omega_i \Omega_i\)

      • \(\omega_i\) = % or weight of portfolio invested in option i
      • \(\Omega_i\) = elasticity of option i
      • So it’s just a weighted average of all the elasticities

• Option Volatility (\(\sigma_{option}\))

  • \(\sigma_{option} = \sigma_{stock} |\Omega|\)
    • Option volatility is higher than stock volatility

• Sharpe Ratio (\(\phi\))

  • \(\phi = \frac{Risk~Premium}{Volatility}\)

    • \(\phi_{Stock} = \frac{\alpha - r}{\sigma_{stock}}\)
    • \(\phi_{Option} = \frac{\gamma - r}{\sigma_{option}} = \frac{\Omega}{|\Omega|} \phi_{stock}\)
      • Last equation basically just means to take direction of \(\Omega\) (+/-) and multiply that by \(\phi_{stock}\)

  • We want the sharpe ratio to be high because we want a high return for low volatility

• Risk Premium

  • Risk Premium on a Stock = \(\alpha - r\)
    • \(\alpha\) = Expected rate of return on a stock
  • Risk Premium of an Option = \(\gamma - r = (\alpha - r)\Omega\)
    • \(\gamma = \Omega\alpha + (1-\Omega)r\) = Expected return of an option

About

• Delta hedging might be the only thing actuaries actually use from IFM material
• Delta hedging is offsetting an options position by buying/selling \(\Delta\) shares of stock
• Market Maker is the person who sells the options contract

Delta Gamma Theta (\(\Delta\Gamma\theta\)) approximations

• \(C_{t+h}(S+\epsilon)\approx C_t(S) + \Delta\epsilon + .5\Gamma\epsilon^2 + \theta h\)
  • A way to approximate the change in options price given the change in a stock price
  • Only use part of the formula for however much they want you to approximate

Marking to Market

• Market maker adjusts his portfolio every day to stay hedged against market swings
• Best demonstrated with an example: Customer buys a t=91/365 day call option, so market maker sells the call option. Suppose S = $40, K = $40, \(\sigma\) = .3, \(r\) = .08, \(\delta\) = 0, and call contract is for 100 shares.
  • Using Black Scholes framework and Greek Formulas we obtain these values in respect to the Call buyer:
    • C(40) = 2.7804, \(\Delta\) = .5824, \(\Gamma\) = .0652, \(\theta\) = -.0173
    • For the market maker, multiply each of these by -1 b/c we are selling the call
      • C(40) = -2.7804, \(\Delta\) = -.5824, \(\Gamma\) = -.0652, \(\theta\) = .0173
  • To hedge, the market maker buys .5824 shares of stock for every call
  • Day 0:
    • Market maker sells option contract so receives premium
      • 2.7804*100 = 278.04
    • Market maker buys \(\Delta\) shares for every option to hedge and spends:
      • -.5824(100)(40) = -2329.6
    • Market maker net position at t=0
      • 278.04 - 2329.6 = -2051.56
      • B/c it’s negative, we must borrow this money and will pay interest on it. If it were +, we would be lending this money and gaining interest
  • Day 1: Stock rises to $40.50. Using Black Scholes w/ t=90/365, C(40) => $3.0621
    • Market maker profit on shares he owns
      • (40.50 - 40)(58.24) = 29.12
    • Market maker profit on change in option value
      • (2.7804 - 3.0621)(100) = -28.17
    • Interest expense market maker must pay. We take previous period balance and compound by rfr
      • \(-2051.56(e^{.08(\frac{1}{365})}-1) = -0.45\)
    • Market Maker’s overnight profit:
      • \(29.12 + -28.17 + -0.45 = 0.5\)
    • Market maker must rebalance his portfolio
      • Use Black Scholes to calculate new \(\Delta\) to be => .6142. So we must buy additional shares:
        • (.6142 - .5824)(100) = 3.18
      • So we must buy 3.18 more shares at new $40.50 price to rebalance portfolio: = 3.18(40.5) = 128.79
      • This isn’t used in the profit equation for the day though
    • Market maker’s new net position at Day 1:
      • (3.0621)(100 options we sold) - (40.5)(61.42 shares we paid for) = -2181.3
  • Day 2: Stock falls to $39.25, t=89/365, using Black Scholes C(40) => 2.3282
    • Market maker profit on 61.42 shares he now owns
      • (39.25 - 40.5)(61.42) = -76.775
    • Market maker profit on option value change
      • (3.0621 - 2.3282)(100) = 73.39
    • Market maker pays interest on previous loan balance
      • \(-2181.3(e^{.08/365}-1) = -0.48\)
    • Market maker’s overnight profit:
      • \(-76.775 + 73.39 + -0.48 = -3.865\)
  • Here’s a summary table of what happens each day:

• Steps in Marking to Market
  • Find market maker’s net Delta position, and offset by buying/selling that many shares
  • Find market maker’s net position from shares and option premiums
  • For the next day
    • Use new Stock price and t in the Black Scholes framework to find new Delta and Call premium
    • Find the profits for your stock and option premium positions
    • Find profit for how much you pay/receive in interest from previous position
    • Add/subtract shares from portfolio to mark to market (rebalance portfolio) w/ new Delta
    • Find new Net position
  • Remember, interest profit never gets added to the net portfolio value

Gamma Hedging

• To stay even more hedged, not only do we make our net \(\Delta\) of our position = 0, but we also make our net \(\Gamma\) of our position = 0.
• We do this by buying/selling shares of another call option to get a net \(\Gamma\) of 0, but then have to buy/sell stock shares to get a net \(\Delta\) of 0
• We have 2 options w/ strikes: \(K_1\) and \(K_2\), each w/ their own \(\Delta\)’s and \(\Gamma\)’s
  • If we sell the \(C(K_1)\) option, we \(\Gamma\) hedge by buying \(\frac{\Gamma_1}{\Gamma_2}\) \(K_2\) Call options to make our net \(\Gamma\) 0
  • Our combined \(\Delta\) becomes:
    • \(\Delta_{combined} = -\Delta_1+\frac{\Gamma_1}{\Gamma_2}\Delta_2\)
    • You can logic your way to this equation if you just remember to buy \(\frac{\Gamma_1}{\Gamma_2}\) contracts of option 2
  • So if this is negative we purchase that many shares to offset our position

Asian Options

• Deals with the average price of the stock
• 2 different kinds of averages
  • Arithmetic Average A(T) = just regular mean
  • Geometric Mean G(T) = \((S_1*S_2*S_3*S_4*...)^{1/N}\)
    • Do not use the initial stock price in calculating the averages, only future ones
• Some notes about the averages:
  • Price of Asian Option \(\leq\) price of equivalent European option
    • To remember this, assume that if you buy an option you know which direction the stock is moving, and the end price will be better than the average.
  • G(T) \(\leq\) A(T)
  • More sampling -> Lower volatility -> Decreases the average
    • So daily stock prices will have a lower average than weekly stock prices
• Recall that payoffs are calculated:
  • Call payoff = Max{0, S-K}
  • Put payoff = Max{0, K-S}
• For Asian options, we replace either S, or K using either A(T) or G(T)
  • Between Call/Put, S/K, A(T)/G(T) there are 8 diff combinations
  • Probs will say “avg price” for replacing S, or “avg strike” for replacing K
  • Probs often have you construct a binomial tree, calculate payoffs at the end of each node, and multiply by the probability of each end node to price the option

Barrier Options

• Types:

  • Knock-In/Out (Up and In / Down and Out) = Option goes in/out of existence if at any point between time 0 and T the stock crosses the barrier once
  • Rebate option: Pays a fixed amount if the barrier is reached
    • Down Rebate = If barrier is below the current price
    • Up Rebate = If barrier is above the current price
• Pricing
  • On IFM, may have to build a binomial tree to solve problems
  • In real life, these are priced using computer simulations
  • Barrier Price \(\leq\) Ordinary Option
    • b/c more restrictive
• Parity relationship for option premiums
  • Knock-in + Knock-out = Ordinary option

Compound Options

• Option to buy an Option
• Have 2 strikes and 2 expirations
• \(t_0\) is current time. At \(t_1\), you have the option to buy (for a price of \(\$x\)) a European option with a strike price of \(\$K\). At time \(T\), the underlying European option expires.
• \(S^*\) is the critical value that if the stock rises above this value, then it makes sense to exercise the compound option.

• Compound Option Parity

  • CallonCall - PutonCall = Call - \(xe^{-rt_1}\)
  • CallonPut - PutonPut = Put - \(xe^{-rt_1}\)
• Problems may have you use Black Scholes to value the underlying option at time t
• For profits, make sure you discount all the cash flows to t=0
  • How much you paid for the compound option at t=0
  • How much you’ll earn/spend at t=t when you exercise the compound option to buy/sell the underlying option
  • The payoff of the underlying option at t=T

Gap Options

• Gap options have a regular strike price \(K_1\) that you buy/sell the asset for, but you can only do so if the stock is more/less than the “trigger” price \(K_2\) at expiration
• Use the same Black Scholes formulas except we change the \(d_1\) formula to have \(K_2\) in place of \(K_1\)
• \(C = Se^{-\delta t}N(d_1)-K_1e^{-rt}N(d_2)\)
• \(P = K_1e^{-rt}N(-d_2) - Se^{-\delta t}N(-d_1)\)
  • \(d_1 = \frac{log(\frac{Se^{-\delta t}}{K_2e^{-rt}}) + .5\sigma^2t}{\sigma\sqrt{t}} = \frac{log(\frac{S_0}{K_2})+(r-\delta+.5\sigma^2)T}{\sigma \sqrt{T}}\)
  • \(d_2 = d_1 - \sigma\sqrt{t}\)
• Put-call parity for gap options:
  • GapCall - GapPut = \(S_0 e^{-\delta T} - K_1 e^{-rT}\)
• Basically everything is the same for Gap options except that we use the barrier/gap \(K_2\) in the \(d\) equations
• Gap options are unique in that they are actually not optional. They must be exercised if the stock price moves and stays past the trigger price. Thus a negative payoff and negative premiums are possible.

Exchange Options

• AKA Outperformance Options
• Basically an option to exchange an asset with another
• S is the price of risky asset 1, K is the price of risky asset 2
• We are given dividend yields (\(\delta_S\) & \(\delta_K\)), volatilites (\(\sigma_S\) & \(\sigma_K\)), and correlation between the 2 continuously compounded dividend yields (\(\rho\))
• Call option is giving asset K (2), and receiving asset S (1). (Paying strike K, receiving stock S) (Same as normal). Put is giving asset S and receiving K.
• Call Prem = \(Se^{-\delta t}N(d_1) - Ke^{-rt}N(d_2)\)
  • \(d_1 = \frac{log(\frac{Se^{-\delta_S t}}{Ke^{-\delta_K t}}) + .5\sigma^2t}{\sigma\sqrt{t}}\)
  • \(d_2 = d_1 - \sigma\sqrt{t}\)
  • \(\sigma = \sqrt{\sigma^2_S + \sigma^2_K -2\rho\sigma_S \sigma_K}\)

Even more exotic options

Following Lognormality, we get these equations:

• \(E[S_t]=S_0e^{(\alpha-\delta)t}\)

• \(Var(S_t) = E(S_t)^2(e^{\sigma^2 t}-1)\)

• \(Med[S_t]=S_0e^{(\alpha-\delta-.5\sigma^2)t}=E[S_t]e^{-.5\sigma^2 t}\)

• \(E[S_t|S_t>K]=S_0e^{(\alpha-\delta)t}\frac{N(d_1)}{N(d_2)}\)
• \(E[S_t|S_t<K]=S_0e^{(\alpha-\delta)t}\frac{N(-d_1)}{N(-d_2)}\)
• \(P(S>K)=N(d_2)\)
• \(P(S<K)=N(-d_2)\)
  • \(d_1 = \frac{log(\frac{S_0}{K})+(\alpha-\delta+.5\sigma^2)t}{\sigma\sqrt{t}}\)
    • Assuming no arbitrage \(\alpha = r\)
  • \(d_2 = d_1 - \sigma\sqrt{t} = \frac{log(\frac{S_0}{K})+(r-\delta-.5\sigma^2)t}{\sigma\sqrt{t}}\)
    • Only diff is we subtract \(.5\sigma^2\) in the numerator

Fitting Stockprices to a Lognormal Distribution

• Exam might give you a table of stock prices over time, and you’ll need to compute parameters

• Historical Volatility (\(\sigma\)) is calculated in 3 steps

  1. Find log ratios
     • \(log(\frac{S_t}{S_{t-h}})\)
  2. Find unbiased estimator of the variance of these ratios
     • \(S^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2\)
     • Or just plug values into multiview
  3. Multiply by 1/h and square root to get SD
     • \(\sigma = \sqrt{S^2/h}\)
       • (For weekly values, h = 1/52, so we’d multiply it by 52)

• Volatility on a stock (another way to get it if they give you a formula)

  • \(\sigma = \sqrt{\frac{Var(Ln(S_t))}{t}}\)

  • Ex: They give that: \(Var(Ln(F^P_{t,0.5}(S)))=.09t\)
    • \(\sigma = \sqrt{\frac{.09t}{t}}=\sqrt{.09}=.3\)
• Computing \(\hat{\alpha}\), (Continuously compounded expected return)
  • Recall that \(ln(\frac{S_{t+h}}{S_t}) \tilde{} N(m = (\alpha - \delta - .5\sigma^2)h, v^2 = \sigma^2 h)\)
    • Once you find all the log ratios, compute the average log ratio (\(\bar{x}\) from the multiview), and this will give you \(\bar{r}\) which equals \(m\) in the above.
    • Use the steps above to compute the sample volatility \(\hat{\sigma}\) using those log ratios
    • For \(h\), use the step size, so if you have monthly stock prices: multiply by \(h=\frac{1}{12}\)
      • (Note, this is different from computing \(\hat{\sigma}\), where we divide by \(\frac{1}{h}\), so we end up multiplying by 12)
    • Once you have those pieces, you can solve for \(\alpha\), but this will be \(\hat\alpha\) b/c we are using historical prices and thus is an estimate.

Variance/Covariance/Correlation Review

• \(Var(X) = E[(X-E(X))^2] = E(X^2)-E(X)^2\)
• \(S^2 = \frac{\sum (x_i -\bar{x})^2}{N-1}\)
  • Unbiased estimator of the variance. Use when dealing w/ a sample
• \(\sigma^2 = \frac{\sum (x_i -\bar{x})^2}{N}\)
  • Actual variance. Use when the population mean is known
• \(Var(aX+bY) = a^2Var(X)+b^2Var(Y)+2abCov(X,Y)\)
  • Cov = 0, when X and Y are independent
  • Cov is usually given on the exam
• \(Var(X+Y+Z) = Var(X)+Var(Y)+Var(Z)+2Cov(X,Y)+2Cov(X,Z)+2Cov(Y,Z)\)
  • Variance of each, plus covariance of every combination with combinatorics factor (2)
• \(Cov(X,Y)=\frac{\sum (x_i -\bar{x})(y_i -\bar{y})}{N-1}\)
  • This is the unbiased estimator, use only N in the denominator for actual covariance
• \(Cov(X,Y) = E(XY)-E(X)E(Y)\)
• \(Cov[aX+bY,cP+dQ]=Cov[aX,cP]+Cov[aX,dQ]+Cov[bY,cP]+Cov[bY,dQ]=acCov[X,P]+adCov[X,Q]+bcCov[Y,P]+bdCov[Y,Q]\)
• \(\rho=Corr(X,Y)=\frac{Cov(X,Y)}{SD(X)SD(Y)}\)

Risk and Return

Mean Variance Portfolio Theory

• M-V Portfolio Theory = Says that investors can evaluate the risk/return of an investment based on expected returns. We use this to find the optimal allocation of assets.

  • Assumptions:
    • All investors are risk averse (prefer less risk for return)
    • Expected return, variances, and covariances of all assets are known
    • To determine the optimal portfolio, we only need to know the above 3 measures
    • There are no transaction costs or taxes

Capital Asset Pricing Model (CAPM)

• \(r_i = E[R_i] = r_f + \beta_i(E[R_{mkt}-r_f)\)
  • \(r_i\) = Required return
  • \(E[R_i]\) = Expected return
  • \(E[R_{mkt}]-r_f\) = Mkt Risk premium, AKA Expected excess return of the mkt
  • \(\beta_i(E[R_{mkt}-r_f)\) = Risk premium for security i, AKA expected excess return of security i
• Notice how we set the required return equal to the expected return. The CAPM is really used to find the required return of asset i, but if the market portfolio is efficient, then this will also be the expected return of asset i.

Cost of Capital

Value of a Firm

• \(V = D + E\)
  • Value of a firm = Debt + Equity

Arbitrage Pricing Theory (APT)

• A multi-factor model. Considers a collection of well diversified portfolios to find the expected return of an asset.
• Similar to the CAPM, but assumptions are not so restrictive
• Assumptions:
  • Uses multiple risk factors, and can use whichever you want (Oil price factor, interest rate, market portfolio factor, etc.)
  • Well diversified portfolios can be created to eliminate non-systematic risk.
• Basically, APT combines a bunch of well diversified portfolios (called a “Factor Portfolio”) and uses all of them to form an efficient portfolio.
• \(E[R_i] = r_f + \sum \beta_i^{F_n}(E[R_{F_n}]-r_f)\)
  • \(E[R_i]\) = Expected return of the asset portfolio
  • \(\beta_i^{F_n}\) = Factor betas of asset i, measures sensitivity of the asset to a particular factor.
  • \(R_{F_n}\) = The return on factor portfolio n
• If they say that hr portfolio is “Self Financing” we rewrite this as:
  • \(E[R_i] = r_f + \sum \beta_i^{F_n} E[R_{F_n}]\)
  • This means we are shorting the risk free asset to fund the asset portfolio
• Fama-French-Carhart (FFC)
  • The most commonly used multi-factor model. Research shows it’s better than the CAPM
  • Considers 4 factors: Market, Market Cap, Book-to-Market Ratios, and Momentum
    • Each of these 4 factors are considered “Self-Financing” (meaning we borrow from something else and invest in the other)
  • \(E[R_i]=r_f + \beta_i^{Mkt}(E[R_{mkt}]-r_f) + \beta_i^{SMB}E[R_{SMB}] + \beta_i^{HML}E[R_{HML}] + \beta_i^{PR1YR}E[R_{PR1YR}]\)
    • Mkt = Market portfolio
    • SMB = Small-minus-big portfolio
    • HML = High-minus-low portfolio
    • PR1YR = Prior 1-year momentum portfolio
  • Self Financing Mkt Portfolio
    • Expected return = \(E[R_{Mkt}] - r_f\)
    • Same as the CAPM model. Accounts for Equity risk. Short the rfr and long the mkt portfolio.
  • Small-minus-big (SMB) Portfolio
    • \(E[R_{SMB}] = E[R_{small}] - E[R_{big}]\)
    • Buys small firms, and finances by shorting big firms. Historically this has been positive b/c small cap tends to outperform large cap.
  • High-minus-low (HML) Portfolio
    • \(E[R_{HML}] = E[R_{HBM}] - E[R_{LBM}]\)
    • Buys high book-to-market ratio stocks and shorts low book-to-market stocks
    • Book value = accounting value of the firm’s balance sheet, how much would be left in assets if the firm went out of business today
    • Market value = price investors are willing to pay in the market for a share of stock. It reflects supply and demand
    • Book-to-Market Ratio = (Book value of firm) / (Mkt Value of firm)
      • If > 1: then the actual worth exceeds the mkt value (Undervalued) (means it’s a value stock)
      • If < 1: then the actual worth is below the mkt value (Overvalued) (means it’s a growth stock)
  • Prior 1-year Momentum (PR1YR) Portfolio
    • \(E[R_{PR1YR}] = E[R_W] - E[R_L]\)
      • \(R_W\) = Last year’s winners
      • \(R_L\) = Last year’s losers
    • This factor accounts for momentum. It buys the top 30% stocks, and shorts the bottom 30% stocks.
• Fama-French (No Carhart)
  • Same as the FCC, but we drop the momentum factor
  • \(E[R_i]=r_f + \beta_i^{Mkt}(E[R_{mkt}]-r_f) + \beta_i^{SMB}E[R_{SMB}] + \beta_i^{HML}E[R_{HML}]\)

Efficient Market Hypothesis

• Past Market Data = Current security prices already reflect all historical stock prices
• Public Information = Current security prices already reflect info from press

• Private Information = Current security prices already reflect info from inside private information and in-depth security analysis

• EMH says you can’t beat the market by knowing the above things because current mkt prices already reflect that information. But people vary in how much they believe it.

• For the exam, you may need to read a study and determine which form of the EMH the study supports or negates.

• Reasons the Market Portfolio is Not Efficient:
  • Not all investors have homogeneous expectations (diff risk preferences and information)
  • Investors are irrational (and misinterpret info)
  • Investors carry portfolios that are mean-variance inefficient (don’t invest on the efficient frontier)
  • Proxy Error = The true market portfolio is every single asset tradable, it’s hard to find a fund like this so we use a proxy like the S&P 500, and there is some error.
  • Other behavioral biases
  • Non-tradable wealth = If you work in a certain industry, and your industry goes down, your human capital in that industry is non-tradable

Investment Risk

Behavioral Biases

Pre-Post money valuation

• Pre-Money Valuation = value of a firm before a funding round
• Post-Money Valuation = value of a firm after a funding round
  • Post-Money Valuation = (Pre-Money Valuation) + (Amount invested)
  • Post-Money Valuation = (Number of shares After the funding round)*(Pre-Money price per share)
• Percentage Ownership: 3 Formulas:
  • = \(\frac{Amount~Invested}{Post-Money~Valuation}\)
  • = \(\frac{(\#~Shares~Owned)*(Pre-Money~Price~per~Share)}{Post-Money~Valuation}\)
  • = \(\frac{\#~of~Shared~Owned}{Total~\#~of~Shares}\)

Capital Structure

• A lot of reading to do in this section. Best to reference the text (Berk and DeMarzo). Or look at last few pages of CA study guide.